Mixing wave-based computing

ABSTRACT

A passive photonics wave-based computing system configured for performing wave-based computing, the system comprising a mixing unit comprising a plurality of inputs and a plurality of outputs the mixing unit being a cavity and further being configured for reflecting or scattering, on average at least three times in the cavity, an optical beam received via said at least one input towards at least one of said plurality of outputs.

FIELD OF THE INVENTION

The invention relates to the field of optical wave-based computing. More specifically it relates to methods and systems for optical wave-based computing with low loss using a mixing unit.

BACKGROUND OF THE INVENTION

The more we get swamped by Big Data, the more we let computers take over the data processing. Machine Learning is therefore currently one of the fastest growing disciplines in computer science and statistics. However, most of this processing still happens in software on power-hungry computers.

An alternative hardware implementation is a so-called Reservoir Computer specifically designed for efficient optical computation. Reservoir Computing is a machine learning branch focusing on processing of time-dependent data. It was first proposed in the early 2000s as a way of using an untrained neural network with internal feedback combined with a trained linear readout layer to perform classification of temporal data. The feedback through the nonlinear nodes in the so-called recurrent network performs a nonlinear mixing of the signal and provides a fading memory to the system.

A passive photonic reservoir computer based on the same reservoir computing principle was already proposed in 2014 by Kristof Vandoorne et al. in Nature communications 5 (2014). However, the proposed implementation suffers from fundamental scaling limits coming from the 3 dB loss associated with the sequence of combiners used in the system.

Passive photonic reservoirs may be used for a variety of applications. Nevertheless, the complexity of tasks the reservoir can perform depends on the number of neurons it consists of, these losses also limit the complexity of tasks that can be performed, and consequently there is still room for improvement.

SUMMARY OF THE INVENTION

It is an object of embodiments of the present invention to provide efficient passive photonic wave-based computing methods and systems based on a mixing unit which show low losses.

The present invention relates to a passive photonics wave-based computing system configured for performing wave-based computing, the system comprising a mixing unit being a cavity comprising at least one input and a plurality of outputs, the mixing unit further being configured for reflecting or scattering of an optical beam, received via said at least one input towards at least one of said plurality of outputs, on average at least three times in the cavity. The mixing element may be a photonic crystal cavity or may be a scattering based mixing unit. It is to be noted that the mixing unit typically may be substantially larger than the wavelength or than the order of the wavelength. The mixing unit may be at least larger than 5 times the wavelength with which the system will be used.

The at least one input may be a plurality of inputs. The mixing unit has a plurality of outputs. Such outputs may be combined, e.g. using weighing and summing, such that the overall wave-based computing system may only have a single output.

It is an advantage of embodiments of the present invention that the mixing unit allows to provide good mixing of input radiation, without large losses occurring. The latter is obtained by avoiding at least some combiners and splitters in the system and instead using a mixing unit.

Where in embodiments of the present invention reference is made to a cavity, reference is made to a region in the photonics device, that is typically filled with transparent materials, being transparent for the wavelengths of the radiation for which the system can be used and that comprises reflective elements at the edge, e.g. reflective walls.

The mixing unit may be configured in the passive photonics wave-based computing system so that it acts as a fading memory for the radiation signals.

The mixing unit may be a cavity comprising a plurality of reflective walls such that after a plurality of reflections radiation from the inputs eventually will reach an output port.

The mixing unit may be a photonic crystal cavity.

The mixing unit may have a shape of a quarter-stadium.

The input and/or output waveguides may be connected to the mixing unit through adiabatic tapers.

The mixing unit may be a cavity made in silicon, whereby ferroelectric thin films are coated on the cavity, wherein the mixing unit is a photonic crystal cavity made of silicon rods wherein in the middle nonlinear polymers are introduced, or wherein the cavity is made in a III-V material.

The system may comprise a plurality of mixing units coupled in a hierarchical arrangement. The cavity may be a photonic crystal cavity made out of silicon with air holes.

The mixing units being coupled in a hierarchical arrangement may for example comprise a first mixing unit taking care of lower-level features in the input signal and the further cavity(ies) for taking care of higher-level features.

The mixing unit may comprise a random set of scattering objects.

The scattering objects may be scattering pillars.

The random set of scattering objects may be positioned in a cavity.

The cavity may be formed from silicon nitride and the scattering pillars may be made of silica.

The present invention also relates to a method for performing photonic wave-based computing, the method comprising applying an optical input signal to one or more inputs of a mixing unit being a cavity, allowing the optical signal to propagate in the mixing unit, said propagating comprising reflecting or scattering of said optical signal received via said at least one input towards at least one of a plurality of outputs, on average at least three times in the cavity, and thus obtaining a nonlinear readout in at least one of said plurality of outputs. The nonlinear readout is obtained after being detected by the detector. In at least some embodiments of the present invention, the method performed is strictly speaking not a traditional reservoir computing technique but rather a sort of spatial variant of the concept, since the inputs, are static. In embodiments of the present invention, typically no time-dependent signals are used.

The method furthermore may comprise sampling the signals at discrete time intervals to combine the signals in a single output signal.

Particular and preferred aspects of the invention are set out in the accompanying independent and dependent claims. Features from the dependent claims may be combined with features of the independent claims and with features of other dependent claims as appropriate and not merely as explicitly set out in the claims.

These and other aspects of the invention will be apparent from and elucidated with reference to the embodiment(s) described hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a mixing unit comprising a photonic crystal cavity with a plurality of input and output connectors, according to an embodiment of the present invention.

FIG. 2a and FIG. 2b illustrate the pulse decay for a cavity with 9 respectively 5 connected waveguides, according to embodiments of the present invention.

FIG. 3 illustrates the harmonical decay of the Q factor and T_(1/2) with increasing number of exit waveguides, according to embodiments of the present invention.

FIG. 4 illustrates sampling of the detected signal at discrete time intervals according to embodiments of the present invention.

FIG. 5 illustrates a mixing unit comprising a plurality of scatterers according to an embodiment of the present invention.

FIGS. 6a to 6c illustrates the effect of scatterers on an incoming radiation signal, according to an exemplary embodiment of the present invention.

FIG. 7 illustrates the BER as function of the delay.

FIG. 8a describes the error for 9 connected waveguides vs. 5 connected waveguides. FIG. 8b describes the target and predicted bit stream at 100 Gps and for the optimal delay (1.55 bit periods) for a SNR=3.

FIG. 9a illustrates that the wave-based computing system can distinguish up to 4 bits without error and FIG. 9b illustrates that for a header of length of 5 bits, the average error rate per header is still well below 5%, illustrating advantages of embodiments of the present invention.

FIG. 10 illustrates a Linear Discriminant Analysis used to discern different headers, as can be used in embodiments of the present invention.

FIG. 11 shows a photonic crystal cavity with connected waveguides according to an embodiment of the present invention. (a) Snapshot of the field profile in the cavity. The mixing of the signal can clearly be witnessed by inspecting the field profiles. (b) Close up of the entrance waveguide. Some scattering losses can be observed at the waveguide-W1 transition.

FIG. 12 shows a block diagram of the simulation in an example according to an embodiment of the present invention.

FIG. 13 shows waveforms FIG. 13(a) detected at two of the exit waveguides as the result of a certain 50 Gbps bit sequence input illustrating features of an embodiment of the present invention. The outputs get sampled once per bit period. It also shows, after the readout, that the prediction approximates the desired XOR target FIG. 13(b). The prediction and the target were aligned by shifting the prediction backwards in time according to the optimal latency of 0.8 bit periods.

FIG. 14 shows that the accumulated power exiting the cavity via the output waveguides adds up to about 83% of the total energy introduced in the system, corresponding to 0.8 dB loss, illustrating features of an example according to an embodiment of the present invention.

FIG. 15 shows FIG. 15(a), the XOR, and FIG. 15(b), AND, of two neighboring bits at 50 Gbps can easily be performed by the reservoir according to an embodiment of the present invention. The latency represents the time shift with respect to the input, after which the output should give the desired value. The correct result of b_(n) XOR b_(n−1) can be retained until a new bit is sent in, while the AND of two bits can be retained until 2 new bits are sent in. Since the BER is cropped at 10⁻³, it can be seen how well the output continuously approximates the target function by looking at the Mean Squared Error (MSE).

FIG. 16 shows for an example according to an embodiment of the present invention that by performing the performance vs latency sweep for every bitrate and looking at the minimal error for each sweep, the operating range of the reservoir can be determined. FIG. 16(a), the optimal bitrate—looking at the MSE—for the XOR task lies at 50 Gbps. However, there is a full band of frequencies the reservoir can work with between 25 Gbps and 67 Gbps. FIG. 16(b), the fact that the linear AND task is easier is again reflected in the the region of operation for the AND, which starts at lower bitrates.

FIG. 17 shows for an example according to an embodiment of the present invention the error Rate (ER) for the worst performing header at each latency. The reservoir can distinguish headers of up to L=6 bits without error at the optimal bitrate of 50 Gbps. To reduce simulation times, the sweep over the latencies was stopped when the ER became higher than 10⁻¹.

FIG. 18 shows for an example according to an embodiment of the present invention the visualization of the separation of three-bit headers by projecting on the two primary LDA axes. A nice separation can be seen for all different headers, while similar headers are located closer together.

FIG. 19 shows for an example according to an embodiment of the present invention that by sweeping over the bitrate to find the operation range, one ca find that the reservoir can distinguish headers up to a header length of L=6 bits without error at a bitrate of up to 100 Gbps.

FIG. 20 shows for an example according to an embodiment of the present invention the decay of the power inside the cavity whereby the envelope decays with a half-life of 10 ps yielding a Q factor of 16400.

FIG. 21 shows for an example according to an embodiment of the present invention that the Q factor decays harmonically with the number of connected waveguides.

Any reference signs in the claims shall not be construed as limiting the scope.

In the different drawings, the same reference signs refer to the same or analogous elements.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The present invention will be described with respect to particular embodiments and with reference to certain drawings, but the invention is not limited thereto but only by the claims. The drawings described are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. The dimensions and the relative dimensions do not correspond to actual reductions to practice of the invention.

Furthermore, the terms first, second and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequence, either temporally, spatially, in ranking or in any other manner. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other sequences than described or illustrated herein.

Moreover, the terms top, under and the like in the description and the claims are used for descriptive purposes and not necessarily for describing relative positions. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other orientations than described or illustrated herein.

It is to be noticed that the term “comprising”, used in the claims, should not be interpreted as being restricted to the means listed thereafter; it does not exclude other elements or steps. It is thus to be interpreted as specifying the presence of the stated features, integers, steps or components as referred to, but does not preclude the presence or addition of one or more other features, integers, steps or components, or groups thereof. Thus, the scope of the expression “a device comprising means A and B” should not be limited to devices consisting only of components A and B. It means that with respect to the present invention, the only relevant components of the device are A and B.

Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment but may. Furthermore, the particular features, structures or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments.

Similarly it should be appreciated that in the description of exemplary embodiments of the invention, various features of the invention are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure and aiding in the understanding of one or more of the various inventive aspects. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed invention requires more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects lie in less than all features of a single foregoing disclosed embodiment. Thus, the claims following the detailed description are hereby expressly incorporated into this detailed description, with each claim standing on its own as a separate embodiment of this invention.

Furthermore, while some embodiments described herein include some, but not other features included in other embodiments, combinations of features of different embodiments are meant to be within the scope of the invention, and form different embodiments, as would be understood by those in the art. For example, in the following claims, any of the claimed embodiments can be used in any combination.

In the description provided herein, numerous specific details are set forth. However, it is understood that embodiments of the invention may be practiced without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.

In a first aspect the present invention relates to a passive photonics wave-based computing system configured for performing wave-based computing. The wave-based computing has similarities with reservoir computing. Nevertheless, use may be made of time independent signals whereas for reservoir computing typically time dependent signals are used. According to embodiments of the present invention, the system comprises a mixing unit being a cavity having at least one input and a plurality of outputs. The cavity may in some embodiments be a space substantially filled with a fluid or a gas. The space may in some embodiments be limited by one or more reflective or scattering walls. The space may in some embodiments be partly filled by scattering or reflective objects, such as for example pillars. It is to be noted that whereas the mixing unit typically has a plurality of outputs, the overall wave-based computing system may have a single output, whereby e.g. the single output of the overall system is obtained by combining, e.g. summing, weighted signals from the outputs of the mixing unit. The mixing unit furthermore is configured for complicated reflecting or scattering of an optical signal received via said at least one input towards at least one of said plurality of outputs. The wave-based computing system may be implemented in a plurality of ways. With complicated or complex reflecting or scattering there is meant that on average a large number of reflections or scatterings are performed before the signal is coupled out. With a large number of reflections, there is meant at least 3, e.g. at least 5, e.g. at least 10 reflections. Where in embodiments of the present invention reference is made to “complex scattering or reflection” or to “on average at least N times the irradiation is scattered or reflected” this may include that on average at least N different portions of the irradiation can be scattered or reflected in different ways (i.e. spatially) and/or that on average a same portion of the irradiation is scattered or reflected at least N times (i.e. temporally). N thereby may be a large number, such as for example 3, 5 or 10. In some embodiments the at least N reflections and/or scatterings may be subsequent in time.

Two particular implementations will be described, one implementation whereby the mixing unit is based on a photonic crystal cavity comprising reflective walls for reflecting in a complex way an optical signal received via at least one input and a second implementation whereby the mixing is based on a plurality of scattering elements, e.g. pillars. Examples of both types of implementations will be described below, embodiments not being limited thereto. The wave-based computing system may, besides a mixing unit comprising at least one input and a plurality of outputs also comprise a readout layer for processing the output of the mixing unit, as well as a further output for outputting a result of the system. The computing system may comprise a non-linearity. The computing system also may comprise a static memory, implemented optically or in a hybrid opto-electronic way.

By way of illustration, an embodiment wherein the mixing unit is based on a photonic crystal cavity comprising reflective walls for reflecting in a complex way an optical signal so as to mix it will further be described with reference to FIG. 1 which shows a schematic drawing of a mixing unit comprising a photonic crystal based cavity together with a plurality of input and output connectors.

A cavity 12 is designed and optimized as the mixing unit 1 of the passive photonics wave-based computing system so that good mixing of the inputs is achieved, e.g. through an effective randomization of the propagating wave direction inside the cavity 12 after a large number of reflections 13. The cavity shape typically displays curved features, for example the arc of a quarter-stadium cavity, but many other cavity shapes may be considered as mixing unit too. In a particular example, a quarter-stadium design may display 30 μm and 15 μm long semi-axis, however these numbers can vary according to a particular task to be solved and the required speed of the input signals. It is an advantage of the present invention to show low losses under the mixing procedure which is achieved by the multiple reflections 13 off the reflective cavity walls 11 before being partially transferred into one or more outputs 14. High reflectivity from the enclosing boundary 11 walls of the cavity may be obtained by the use of a photonic crystal structure comprising a regular lattice of air holes in silicon but is not limited thereto and other material interfaces of the cavity walls may be conceived. Propagation losses, scattering losses, surface losses, and other sources of imperfections in the material and fabrication are typically small compared to the coupling losses induced by the plurality of input and output ports 10 connected to the cavity. Hence the number of input and output connectors 10 constitutes the major control variable of the cavity's quality factor (Q-factor). The Q-factor is related to the lifetime of a cavity 12 excitation through one or more of its input connectors 10; the final decay of any excitation through loss mechanisms provides a form of fading memory to the passive photonic computing system. A reasonable minimum number of input and output connectors to perform computational tasks with low error rates is typically ranging between five and nine, corresponding to Q-factors between 3000 and 5000, but different choices may lead to better results in some cases. A higher Q-factor can be obtained by reducing the number of input and output connectors which gives rise to better system memory but trades system bandwidth.

It is of advantage of the described embodiment that the input connectors 10 can serve as output connectors at the same time, thus limiting the complexity of the overall system. Furthermore, the input or output connectors 10 typically comprise of an adiabatically controlled taper 101 connecting to the external input or output waveguides and a defect waveguide 102 inside the photonic crystal structure, both ensuring that input and output radiation signals are delivered to and coupled out from the cavity 12. The tapered sections 101 can be tailored to inject or extract a given fraction of the incident radiation signal, hence are another means to control the cavity's Q-factor. Typical transmissivity values range from 30% to 95% but can be adapted accordingly and on an individual basis. Moreover, it is possible, although not required to add delay lines to the input or output connectors 10 in order to provide for increased memory and better signal mixing properties of the mixing unit 1.

Furthermore, an embodiment of the present invention may benefit from the flat surface structure of the cavity 12 which is ideally suited for deposition of thin film coatings, e.g. ferroelectric thin films but not limited thereto. In a likely manner, a cavity region 12 based on air surrounded by photonic crystal walls made from silicon rods is an embodiment of the present invention that could be filled with a nonlinear material, nonlinear polymers for instance. Another possibility is to pattern the cavity 12 into a III-IV material. All the mentioned optional features provide an enhanced nonlinear response to the mixing unit 1 which may lead to increased efficiency or computational power of the passive photonic computing system.

It may be an advantage of the present invention to stack several mixing 1 units together or to combine them into a hierarchical architecture which may allow for a more efficient allocation of computing resources, faster problem solving, or the treatment of derived high-level features.

By way of illustration, further features and characteristics will be described with reference to FIG. 1. The particular quarter-stadium shape shown typically shows interesting dynamics and results in complete mixing of the fields, in the sense that an input wave will obtain all possible wave-vectors in the cavity. The mixing unit behaves like a fading memory necessary for wave-based computing.

With respect to its operating, in order to perform operations at a certain bitrate, the lifetime of the cavity should be at least larger than the bit period. Apart from optimising Q-factor and waveguide coupling, one can also simply increase the size of the cavity if one wants to process signals with a lower bitrate. This is a valid option, since the propagation losses are small compared to the scattering and coupling losses.

Another important parameter closely related to the Q-factor of the cavity is the half-life T_(1/2) of a pulse in the cavity, which is deemed as the point where the amplitude of the envelope of the field reaches half of the original amplitude. The half-life is related to the Q factor as

$T_{\frac{1}{2}} = \frac{Q\; {\log (2)}}{\pi \; f}$

By way of illustration FIG. 2a and FIG. 2b illustrate the pulse decay for a cavity as shown in FIG. 2 having 9 connected waveguides respectively 5 connected waveguides. The decay for a pulse having a half-life of 5 ps in the cavity is shown. The half-life is—just as the Q-factor—a determining factor for the memory of the system. A reduced number of connected waveguides (b) yields a higher T_(1/2) (and thus longer memory) of 8.2 ps, while more connected waveguides yield a shorter half lifetime T_(1/2) of 4.42 ps which will result in a shorter memory. FIG. 3 illustrates the Q factor and T_(1/2) decay monotonously with increasing number of exit waveguides.

It is immediately clear that a higher pulse half-life will result in a longer (fading) memory.

Therefore (as is depicted in FIG. 2b and FIG. 3), the memory can be considerably increased by reducing the number of connected waveguides. However, this reduces the number of output nodes that can be used by the readout. There will thus be a clear trade-off between the computing power of the cavity (number of exit waveguides) and the memory of the cavity (Q-factor)³. The trade-off will also very much depend on the application being studied, making it difficult to give concrete guidelines. However, the Q factor should be more than 3000 in order to perform useful operations.

In some embodiments, the cladding is used for further designing the cavity. The cladding may be made of a photonic crystal, but alternatively in one example the whole cavity can in fact be made purely out of silicon with air cladding. However, to reduce losses, a photonic crystal cavity may be preferred. This design can consist out of a silicon slab with air holes or even the reverse option. The first step in designing the cavity preferably comprises optimizing the Q-factor for the desired wavelength range. This optimization can for example be done by performing several FDTD simulations for different values of the lattice constant of the photonic crystal and the radius of the photonic crystal. Next, the entrance efficiency of the entrance coupling may be optimized by optimizing an adiabatic taper where the holes of the photonic crystal are systematically introduced in an adiabatic manner.

In some embodiments, if the intensity of the light is high enough inside the cavity, the nonlinearities of silicon become more and more important. These nonlinearities can yield important extra computing power in the context of wave-based computing. To enhance these inherent nonlinearities in silicon, one could in principle deposit nonlinear thin films on top of the cavity such as the ferroelectric materials LiNbO3 or BaTiO3. Another way is to have a photonic crystal cavity made out of silicon rods, where in the middle (the air cavity) some nonlinear polymers such as rhodamine are introduced. A third option is for example to make the cavity out of known III-V materials, where the inherent nonlinearities are higher than in silicon.

In some embodiments, the signal may be inputted in more than 1 access waveguide. E.g., inputting the original input in 1 waveguide, and a suitably delayed signal in another waveguide will lead to better mixing.

In some embodiments, as a possible solution for the memory-outputs trade-off, use is made of multiple coupled photonic crystal cavities in a hierarchy, whereby the first cavity takes care of lower-level features in the input signal, and the next cavities take care of higher-level features. Alternatively, when needing to detect an 8-bit pattern e.g., there could be one cavity to recognise the first 4 bits, and a second cavity with a suitable delay to detect the last 4 bits. This information could then be combined in a higher-level readout layer in order to detect the full 8 bits.

Also, by way of illustration, an embodiment wherein the mixing unit of the wave-based computing system is based on a plurality of scatterers, optionally in a cavity, is further discussed. An example thereof is shown in FIG. 5, in the present example being suitable for identification of types of cells (e.g. cancer cells) and more particularly by direct processing of a hologram using wave-based computing, although the invention is not limited thereto and also other types of characterization can be performed.

The scatterer configuration of the plurality of scatterers has a large number of degrees of freedom and its complete exploration, even using smart methods as evolutionary algorithms, would be computationally quite expensive. In fact, for each tested configuration, hundreds or thousands of simulations have to be performed in order to provide the classifier with a sufficient number of training and test samples. Therefore, only a few general parameters that control the complexity of the collected interference pattern were explored, looking for a maximum in the classifier performances. In the present example, the scatterers are placed in layers, e.g. vertical layers as shown in FIG. 5 with an average vertical distance between their centers of The center of each scatterer is randomly displaced with respect to their unperturbed center in the layer, both along the vertical and the horizontal directions. All the random displacements are sampled from the same uniform probability distribution. The considered parameters for the structure optimization are:

-   the amplitude A_(rand) of the scatterers' random displacement; -   the horizontal distance D between the layers; -   the number N_(layers) of layers.

An example of how these parameters can modify the interference pattern is given in FIGS. 6a, 6b and 6c . FIG. 6a illustrates the situation without scatterers, FIG. 6b illustrates the situation with 4 layers of scatterers without random displacement and FIG. 6c illustrates the situation of 4 layers of scatterers with random displacement. It can be seen that for this particular incarnation, the interference pattern in case no scatterers are present is relatively simple and smooth and confined between −6° and 6° , that the interference pattern in case of 4 layers of scatterers without random displacement shows peaks only in bounded angle regions between −50° and 50° and that the interference pattern in case of 4 layers of randomly positioned scatterers is complicated and shows intensity peaks between −60° and 60°.

In a second aspect, the present invention provides a method for performing computation on the wave-based photonic system.

The method according to embodiments comprises applying an input to one or several of the connected inputs, e.g. input cavity waveguides. In a following step, the signal then propagates through the mixing unit and mixes in a complicated and possibly non-linear manner. The mixing of the original signal happens in a passive way, until the signal is outputted. For the outputting, a quadratic nonlinearity may be applied the complex signal consisting of an amplitude and phase may result in a real-valued magnitude of the signal. This typically happens when reading out with a photodiode. After the readout step, the readout signal may in some examples be sampled at discrete time intervals to be combined into a single output signal using pretrained weights. The sampling is shown in FIG. 4.

By way of illustration, a particular embodiment based on a mixing unit comprising a photonics waveguide cavity will be described below. The exemplary method comprises the following steps described hereafter, but modifications and extensions to the said steps are not excluded.

One or more input signals are injected into the body of the cavity 12 making use of the adiabatically tapers 101 and defect waveguides 102 of the input connectors 10. The input signals are typically guided, time-encoded, modulated photonic waves that were coupled into the signal carrying waveguides surrounding the mixing unit 1 and terminating at the input connectors 10. Each of the input signals injected into the cavity 12 undergoes a long sequence of reflections 13 off the cavity walls 11, spreading at the same time by the phenomena of diffraction of the aperture at the end of the defect waveguides 102. The reflections 13 and propagation of the excited waves inside the cavity 12 are typically nearly lossless. During the propagation inside the cavity 12, the excited waves encounter 14 pierced sections of the cavity wall, the output connectors 10, the purpose of which is to couple out a fraction of the light wave incident on that connector. The outcoupled waves from the cavity 12 are then routed to a detecting unit, e.g. an integrated photodetector module, which applies a nonlinear readout transformation. Typically, this occurs to be the conversion of photonic guided signals in the complex domain to their electronic counterpart in the real domain only; the nonlinear transformation being the square modulus translating electric field strengths into corresponding optical power values. In a subsequent step, the nonlinearly transformed signal may be sampled at regular time intervals and combined into a single output signal by a linear weighting function.

The second aspect also relates to a method wherein the mixing step is performed by scattering using a plurality of scattering elements, e.g. pillars. The output of the mixed signal may then be led to a detection unit where the output signal may be evaluated.

EXAMPLE 1

In the first example, benchmarking of the wave-based computing system as shown in FIG. 1 is discussed.

A first benchmark for the memory of the signal is the ability to reproduce the exact input with a certain amount of delay. From FIG. 7, for the current design, there is quite a wide basin of operation. Using 5 outputs results in being able to reproduce the input with a larger delay, which makes sense because of the increased Q factor and thus memory for 5 outputs.

A second benchmark which illustrates the ability to perform Boolean operations is the nonlinear xor task, where the xor is taken between two bits b^(n) and b^(n-k), k bits apart. Since a normal conventional linear classifier can only achieve a minimum of 25% error rate, it is also a good performance indicator of the nonlinearity in the system.

First, the performance of an xor on two neighboring bits was assessed, while continuously increasing the bit separation. This is called the xor-specific memory of the reservoir, as it is an indicator of how good the wave-based computing system can remember the combined xor of two bits. As can be seen in FIGS. 8a and 8b , the performance is clearly worse than the delay memory.

Contrary to the linear memory task, the performance of the wave-based computing system does not really depend on the number of waveguides that is being used as input waveguides, as can be seen in FIGS. 8a and 8b .

This is probably because the performance in this case is determined limited by the nonlinear character of the task, and not so much by the linear memory capacity.

In fact, the most important figure to classify the performance of the reservoir, is probably the bitrate at which it operates optimally. The performance of the wave-based computing system for the xor operation between neighboring bits b^(n)xorb^(n−1). This operation is optimal at 45 Gbps for the wave-based computing system with 9 connected waveguides. This bitrate may seem high, but this optimal bitrate can be easily reduced by creating a bigger cavity. Therefore, one also choose to quantify the optimal performance by the dimensionless parameter T_(1/2)/T_(bit). This allows us to easily scale the wave-based computing system operating range by tweaking its design parameters.

As a more useful and general task, the performance of the wave-based computing system in recognizing headers in a bitstream was assessed. The simple cavity performs reasonably well for header recognition tasks, with an errorless recognition of up to 4 bit headers, while very good performance for 5 bits headers is also achievable. Note that performance for a higher number of bits can be improved by a cascaded system, as mentioned above, by training each of the cascaded systems for a different subset of the header. In this case, Linear Discriminant Analysis was used as the linear classification algorithm. This algorithm allows us to project the 9-dimensional (# connected waveguides) output state to a lower dimensional state, as is shown in FIG. 9a , FIG. 9b and FIG. 10.

EXAMPLE 2

By way of illustration, embodiments of the present invention not being limited thereto, a second example is further discussed of use of a photonic crystal cavity with a quarter-stadium shape, which fosters interesting mixing dynamics. These mixing properties turn out to be very useful for memory-dependent optical signal processing tasks, such as header recognition. The proposed, ultra-compact photonic crystal cavity exhibits a memory of up to 6 bits, while simultaneously accepting bitrates in a wide region of operation. Moreover, because of the inherent low losses in a high-Q photonic crystal cavity, the proposed design is very power efficient.

The system consists of an on-chip photonic crystal cavity in the shape of a quarter-stadium resonator, which fosters interesting mixing of the fields in an almost chaotic manner. The photonic crystal cavity was designed for the 220 nm silicon photonics platform, consisting of holes etched from a 220 nm silicon slab with radius r=0.37a, for which a =420 nm. The light is sent through one of the seven standard 450 nm waveguides, which are connected to W1-defects in the wall of the photonic crystal cavity. The light inside the cavity subsequently leaks out of the cavity via via all of those defects. The six other defects are used for readout. Alongside its mixing property, it also possesses a fading memory: the signal is bound to remain in this cavity for a certain amount of time directly proportional to the Q-factor and the dimensions of this cavity as shown in FIG. 12. Moreover, light is trapped inside the cavity and can only leak out using the wall defects, where they contribute to the useful output signal. This results in low overall losses in the reservoir. The size of the cavity will obviously have a profound effect on the region of operation of the reservoir in terms of the bitrate. The dimensions of the cavity with 7 W1-waveguides (FIG. 11(a)) are 30 μm×60 μm and were chosen to exhibit a memory of up to 5 bits at the highest attainable bitrate of the detector (around 25 Gbps). The photonic crystal cavity used allows for an extremely small design resulting in a reservoir with dimensions smaller than 0.01 mm². For these dimensions, the cavity exhibits a Q-factor of approximately 16400, which corresponds to a pulse half-life of about 20 ps. This means that for the requested binary input of 25 Gbps, about 4 to 5 bits will be able to get mixed with each other before the intensity of the first bit drops below the noise level. As will be shown, this results in sufficient memory for the XOR task and up to 5-bit header recognition at 25 Gbps. However, this frequency cutoff is not steep, and the detector remains useful for bitrates up to about 50 Gbps, as will be shown.

To obtain the response of the photonic crystal reservoir to an arbitrary bit stream, the most time-consuming step of the calculation is simulating the propagation of the light through the cavity. However, the propagation of the complete bit stream is not simulated, as this would result in enormous calculation times. Instead, an alternative approach is used illustrated in FIG. 12. Suppose one has a bit streams consisting of the bits b₁, b₂, . . . , b_(N) ∈ {0, 1}, for which the nonzero bits are specified by a pulse u(t):

${{u(t)} = {{\begin{pmatrix} {E_{0}(t)} \\ {H_{0}(t)} \end{pmatrix}\mspace{14mu} {with}\mspace{14mu} {u(t)}} = {{0\mspace{14mu} {if}\mspace{14mu} t} < {0\mspace{14mu} {or}\mspace{14mu} t} > T}}},$

with T the bit period of the signal. Then the value of the bit stream at time t is given by

${x(t)} = {\begin{pmatrix} {E_{in}(t)} \\ {H_{in}(t)} \end{pmatrix} = {{\sum\limits_{n = 1}^{N}{b_{n}{u\left( {t - {nT}} \right)}}} = {b_{k}{u\left( {t - {kT}} \right)}}}}$

Each of the exit waveguides i will have an exponentially decaying response U_(i)(t) to the single bit pulse u(t), with U_(i)(t)>0 for all t>0. This means that the response X_(i)(t) to the total bit stream at waveguide i can be described by

${X_{i}(t)} = {\begin{pmatrix} {E_{out}^{i}(t)} \\ {H_{out}^{i}(t)} \end{pmatrix} = {\sum\limits_{n = 1}^{N}{b_{n}{{U_{i}\left( {t - {nT}} \right)}.}}}}$

Note that in this case, the responses cannot be decoupled anymore since U_(i)(t)≠0 for t>T.

This means that the state of the reservoir depends linearly on the previous input values. In practice, the responses P_(i)(t) at each of the waveguides to a single pulse are recorded from an FDTD simulation, performed by the Lumerical FDTD software. Then the response of a complete bit stream (typically a PRBS signal of 10⁵ bits) is calculated by coherently adding together the individual bit responses for each channel in the above described way. Note that this described procedure is just a “bit-level” version of the impulse response method, where the response of an arbitrary system is found by convolving the function with the response of an ultrashort impulse. Here, it was chosen to work with the “bit-level” response instead of the true impulse response because of numerical rounding errors.

Finally, the photodetector performs nonlinear operation by detecting the light. This detector is simulated by a detector model with similar parameters as the detector in the setup, which has a load resistance R_(L)=1 kΩ, a bandwidth f_(c)=25 GHz and a responsivity η0.5 A/W. The detector noise introduced by the amplification of the photogenerated current is described by white thermal noise I_(tn), modelled as a Nyquist process and shot noise I_(sn), modelled as a Poisson process:

$I_{n} = {\sqrt{I_{in}^{2} + I_{sn}^{2}} = {\sqrt{\left( \frac{4k\; {Tf}_{c}}{R_{L}} \right)^{2} + {2{qIf}_{c}}}.}}$

In the readout, the reservoir output streams are sampled a fixed number of times per bit period, as is shown in FIG. 13(a). After this, a linear combination of the sampled values is made according to weights that are specifically trained for the intended application. Increasing the number of sampling points generally improves the performance of the reservoir. However, this work is already done at very high bitrates, it is usually chosen to sample only once per bit. To obtain the weight matrix W_(out) for the linear combination acting on the output states of the reservoir, two different kinds of training algorithms are used: ridge regression (linear regression with a regularization parameter) for the binary classification tasks such as the XOR task, and linear discriminant analysis (LDA) for multi-class classification tasks such as header recognition. Linear Discriminant Analysis is a well-known linear machine learning technique, seeking to find the best linear combination of features (in the present case output signals) that best separates the different classes of the problem. It is often used for dimensionality reduction of the problem by discarding the least important singular values of the weight matrix.

The reservoir can be trained to reproduce the desired target signal with a certain delay, which is called the latency of the reservoir. The latency is usually expressed as a multiple of the bit period, i.e. the number (or fraction) of bit periods you have to wait after the last relevant bit has completely entered the cavity before you can reproduce the target signal. There usually exists only a narrow range of latencies for which the target can be approximated without error. The approximation for an XOR-target at optimal latency is shown in FIG. 13(b). Note that the target and the prediction is aligned by shifting the prediction backwards in time according to the latency.

The simulated total energy measured at the exits of the cavity is about 83% of the total energy inserted, as can be seen in FIG. 14. Looking deeper into the source of the losses, it is found that there is a slight mode mismatch between the access waveguides and the W1 photonic crystal waveguides. When coupling out the light from the cavity, this causes scattering out of the exit waveguides, resulting in lost power. Further optimization to improve this transmission can still be performed.

In the simulations, a PRBS of 10⁵ bits with an input power of 1 mW is sent through one of the photonic crystal W1-waveguides (the top waveguide on the left in FIG. 11(a)). The responses of the six other waveguides are then recorded. Finally, on the recorded output stream, the readout weights are trained to follow the intended target function with an as low as possible mean squared error, as is show in FIG. 13(b). After performing a threshold, the bit error rate (BER) is calculated. Since 10⁵ bits are used in the simulations, the general guideline is to crop the BER at 10⁻³, i.e. 2 orders of magnitude higher than the lowest BER one can find in the simulation reported in Jeruchim “Techniques for estimating the bit error rate in the simulation of digital communication systems”, J-SAC 2(1) (1984) p 153-170.

As one of the binary target functions, the XOR of two consecutive bits was chosen, which is known in machine learning to be a hard, nonlinear task due to the fact that the output cannot be found by just performing a linear classification algorithm such as linear regression on the inputs. However, as can be seen in FIG. 13(a) and FIG. 15(a), sending the input first through the photonic crystal cavity and training a readout on the nonlinearly detected outputs of the reservoir, clearly helps to perform the XOR.

During the optimization of the readout, the latency is swept. FIG. 15 illustrates again that the nonlinear XOR is a quite difficult task, as the maximal latency with errorless performance is about half of the maximal latency of the AND task.

By performing the performance vs latency sweep for every bitrate and looking at the minimal error for each sweep, one can infer the operation range of the reservoir. Looking at FIG. 16, one can see that this is where this reservoir really excels: by just adjusting the readout weights, this simple cavity can perform the XOR task reliably between 25 Gbps and 67 Gbps. This region of operation is much wider than previously reported values for the XOR task.

At higher bitrates, both the AND and the XOR stop working. This can be explained by the fact that the reservoir remembers too many previous bits, which increases the signal to noise ratio on the most recent bits, which are relevant for the operation. The interesting result here is that it was successful to compute a highly nonlinear function such as XOR by using a completely passive device. This is of course only possible because of the nonlinearity of the photodetector, which takes the magnitude of the complex-valued field at the exits of the reservoir.

For applications in telecom, recognizing headers in the bit stream is often a more useful task.

Interestingly, the exact same simple cavity design can perform this task as well. Concretely, all different headers were searched for simultaneously in a completely random bit stream. For each bit in the bit stream, a class label was given corresponding to the header of length L made by the current bit and the L−1 previous bits. Then, LDA was used to find a different weight matrix for each of the different classes. At each different bitrate, the error rate for the worst performing header (class) is reported.

As can be seen in FIG. 17, errorless recognition of up to 6-bit headers is possible at 50 Gbps. Apart from providing us with readout matrix, LDA also allowed to project the 6-dimensional (number of exit waveguides) reservoir output state onto a lower dimensional state, in which one can see the separation of the headers visually in FIG. 18.

Finally, in terms of bitrate, the simulations show that longer headers work better at lower bitrates, as can be seen in FIG. 19. This is unsurprising, as for longer headers, one needs to keep more bits in memory. Therefore, the bitrate needs to be higher to accommodate this.

In the previous simulations, always 7 connected waveguides were chosen. This choice was rather arbitrary, and one could argue that changing the number of connected waveguides will have a profound effect on the quality of the reservoir, as decreasing the number of exit waveguides will inevitably increase the Q-factor and thus the memory of the reservoir. This form of optimization is far from trivial, as removing exits will likely also decrease the complexity of the tasks the reservoir can solve. To quantify this effect, the performance of the reservoir on the XOR task for a range of number of exits was studied. When the light source is turned off, the power in the cavity with 7 connected waveguides decays exponentially with a slope m =−0,070 ns⁻¹, as can be seen in FIG. 20. This yields for the Q-factor at X=1550 nm:

$Q = {{- \frac{2\pi \; c}{\lambda \; m}} = 16400.}$

A more useful value for the time scale of the reservoir is however the half-life T_(1/2) of the pulse, as it provides a more tangible metric for the memory of the reservoir.

$T_{1/2} = {{- \frac{\log (2)}{m}} = {10\mspace{14mu} {{ps}.}}}$

Looking at the Q-factor and half-life for several variations of the cavity with less connected waveguides, one can see in FIG. 21, as one would expect, that the Q factor decays harmonically with the number of connected waveguides. It can also be clearly seen that the threshold for the number of connected waveguides for the XOR task lies at 6 waveguides (1 input and 5 outputs).

Although certainly possible, increasing the number of connected waveguides even more will not yield better results for the benchmark tasks presented here, however it might prove to be a useful tactic for tackling some more challenging problems in the future. One should keep into account, however, that introducing extra waveguides will also have an adverse effect on the Q-factor of the cavity and thus also on the memory of the reservoir.

The above example shows that the photonic cavity shows excellent performance on Boolean tasks such as the nonlinear XOR and the AND task. Additionally, the exact same reservoir can also perform header recognition for up to 6-bit headers. The simulations also show that this reservoir accepts a very wide range of bitrates, with good performance between 25 Gbps to 67 Gbps for the XOR-task, while the header recognition task can be performed up to 100 Gbps without having to change anything to the design. For both applications, the region of operation can probably still be extended to lower bitrates by increasing the size of the cavity. Conversely, decreasing the size of the cavity should yield better performance at higher bitrates, allowing to trivially upscale the working range of the reservoir to bitrates that are unattainable by most systems in the state of the art. 

1.-15. (canceled)
 16. A passive photonics wave-based computing system configured for performing wave-based computing, the system comprising a mixing unit comprising at least one input and a plurality of outputs, the mixing unit being a cavity further being configured for reflecting or scattering an optical beam, received via said at least one input towards at least one of said plurality of outputs, on average at least three times in the cavity.
 17. The passive photonics wave-based computing system according to claim 16, wherein the mixing unit is configured in the passive photonics wave-based computing system so that it acts as a fading memory for radiation signals, by reflecting or scattering in the cavity a received optical beam on average at least 5 times, for example on average at least 10 times.
 18. The passive photonics wave-based computing system according to claim 16, wherein the mixing unit is a cavity comprising a plurality of reflective walls such that after a plurality of reflections radiation from the inputs eventually will reach an output port.
 19. The passive photonics wave-based computing system according to claim 16, wherein the mixing unit is a photonic crystal cavity.
 20. The passive photonics computing system according to claim 16, wherein the mixing unit has a shape of a quarter-stadium.
 21. The passive photonics wave-based computing system according to claim 16, wherein the input and/or output waveguides are connected to the mixing unit through adiabatic tapers.
 22. The passive photonics wave-based computing system according to claim 16, wherein the mixing unit is a cavity made in silicon, whereby ferroelectric thin films are coated on the cavity or wherein the mixing unit is a photonic crystal cavity made of silicon rods, wherein in the middle nonlinear polymers are introduced, or wherein the cavity is made in a III-V material.
 23. The passive photonics wave-based computing system according to claim 16, the system comprising a plurality of mixing units coupled in a hierarchical arrangement.
 24. The passive photonics wave-based computing system according to claim 16, wherein the mixing unit comprises a random set of scattering objects.
 25. The passive photonics wave-based computing system according to claim 24, wherein the scattering objects are scattering pillars and/or wherein said random set of scattering objects are positioned, during use, in a cavity and/or wherein objects to be recognized are positioned, during use, in a cavity.
 26. The passive photonics wave-based computing system according to claim 24, wherein the scattering objects are positioned in a resonator for allowing that the input signal has multiple passes through the scattering pillars and/or wherein objects to be recognized are positioned in a resonator for allowing that the input signal has multiple passes through the scattering pillars.
 27. The passive photonics wave-based computing system according to claim 26, wherein the cavity and/or propagation region are formed from silicon nitride and wherein the scattering pillars are made of silica.
 28. The passive photonics wave-based computing system according to claim 24, wherein the system comprising a plurality of mixing units coupled in a hierarchical arrangement.
 29. A method for performing photonic wave-based computing, the method comprising: applying an optical input signal to one or more inputs of a mixing unit being a cavity, allowing the optical signal to propagate in the mixing unit, said propagating comprising reflecting or scattering of said optical signal received via said at least one input towards at least one of a plurality of outputs, on average at least three times in the cavity, and thus obtaining a nonlinear readout in at least one of said plurality of outputs.
 30. A method according to claim 29, wherein the method furthermore comprises sampling the signals at discrete time intervals to combine the signals in a single output signal. 